American Options, Multi–armed Bandits, and Optimal Consumption Plans: A Unifying View

Abstract

In this survey, we show that various stochastic optimization problems arising in option theory, in dynamical allocation problems, and in the microeconomic theory of intertemporal consumption choice can all be reduced to the same problem of representing a given stochastic process in terms of running maxima of another process. We describe recent results of Bank and El Karoui (2002) on the general stochastic representation problem, derive results in closed form for Lévy processes and diffusions, present an algorithm for explicit computations, and discuss some applications.

Keywords: American options, Gittins index, multi–armed bandits, optimal consumption plans, optimal stopping, representation theorem, universal exercise signal.

AMS 2000 subject classification. 60G07, 60G40, 60H25, 91B16, 91B28.

Support of Deutsche Forschungsgemeinschaft through SFB 373, ”Quantification and Simulation of Economic Processes”, and DFG-Research Center ”Mathematics for Key Technologies” (FZT 86) is gratefully acknowledged.